Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian. (Q1396413)

Let \(G\) be a compact semisimple Lie group, \(\mathcal G\) its Lie algebra, \((\;,\;)\) an \(\text{Ad}_G\)-invariant inner product on \(\mathcal G\) and \(M\) an adjoint orbit in \(\mathcal G\). In the paper under review the author takes in account the particular case when \((M,(\;,\;)_M)\) is a Kähler manifold with respect to its canonical complex structure, and \(L\) is a closed minimal Lagrangian submanifold of \(M\). In this picture he gives an upper bound of the first eigenvalue of the Laplacian on \(L\). The particular case when \(G=SU(n)\) is also nicely pointed out.